A 5.0 m tree is leaning 5° from the vertical. To prevent it from leaning any farther, a stake needs to be fastened 2 m from the top of the tree at an angle of 60°with the ground. How far from the base of the tree, to the nearest metre, must the stake be?
How did you get 95° for angle b
To solve this problem, we can use trigonometry.
Step 1: Calculate the height of the tree that is leaning.
Using trigonometry, we can use the formula:
Height of the tree = Actual height / Cosine(leaning angle)
Height of the tree = 5.0 m / Cos(5°)
Height of the tree ≈ 4.996 m (approximated to 3 decimal places)
Step 2: Find the distance from the base of the tree to the stake.
Using trigonometry, we can use the formula:
Distance from the base = Height of the tree - (Stake height / Tan(stake angle))
Distance from the base = 4.996 m - (2 m / Tan(60°))
Distance from the base = 4.996 m - (2 m / √3)
Distance from the base ≈ 3.33 m (approximated to 2 decimal places)
Therefore, the stake should be placed approximately 3.33 meters from the base of the tree.
To solve this problem, you can use trigonometry and the concept of right triangles. Here's how you can find the distance from the base of the tree to the stake:
1. Visualize the problem: Draw a diagram of the situation to help understand the given information. Draw a vertical line to represent the tree, and mark a point 5.0 m from the ground level to represent the top of the tree. From this point, draw a line that is 5° from the vertical, representing the lean of the tree. Finally, draw another line from the top of the tree, 2 m long, at an angle of 60° with the ground. This line represents the stake.
2. Identify the right triangle: In your diagram, you should now have a right triangle formed by the vertical line (representing the tree), the line of the lean, and the line of the stake. The point where the line of the stake intersects the vertical line (tree) is the base of this right triangle.
3. Use trigonometry: Recall that in a right triangle, the sine function relates the length of the side opposite an angle to the length of the hypotenuse. In this case, we know the angle and the length of the side opposite it, but we want to find the length of the hypotenuse (distance from the base to the stake).
4. Calculate the length of the hypotenuse: Let's denote the length of the hypotenuse by "d" (the distance from the base to the stake). We can use the sine function to relate the known values:
sin(60°) = opposite / hypotenuse
Since we know that the hypotenuse is 2 m, we can rearrange the formula to solve for the opposite side:
opposite = hypotenuse x sin(60°)
= 2 x sin(60°)
Using a calculator, evaluate sin(60°) and compute the value of the opposite side.
5. Find the distance to the base: Now that we know the length of the opposite side (the distance from the top to the stake), we can subtract that value from the total height of the tree to find the distance from the base. Since the height of the tree is given as 5.0 m, we subtract the length of the opposite side from it:
distance to the base = tree height - opposite side
Substitute the values you obtained into the formula and round to the nearest meter to find the final answer.
Following these steps, you can determine the distance, to the nearest meter, from the base of the tree to the stake.
I visualize a triangle ABC such that
AB = 3
angle B = 95°
angle C = 60° , so angle A = 180-95-60 = 25°
BC = x, our required length
By the sine law:
x/sin25 = 3/sin60
x = 3sin25/sin60 = appr 1.46 m